Question

How do you simplify ${{\left( {{c}^{2}}{{k}^{5}} \right)}^{4}}$ ?

Answer 1

Hint: For these kinds of questions, all we need to know are the law of exponents. Before solving these questions, let us check which law of exponent can help us simplify this. We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}={{a}^{mn}}$ . Let us compare and see what our $a,m,n$ are in this question and solve by using this particular law of exponents. We should also distribute the power to our both variables namely $c,k$ as it is multiplied for both.

Complete step by step solution:
The law of exponents that we going to use to simplify this question is ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}={{a}^{mn}}$. In the question, we are given ${{\left( {{c}^{2}}{{k}^{5}} \right)}^{4}}$.
So now before applying the law, let us distribute the power to both the variables.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow {{\left( {{c}^{2}}{{k}^{5}} \right)}^{4}} \\
 & \Rightarrow {{\left( {{c}^{2}} \right)}^{4}}{{\left( {{k}^{5}} \right)}^{4}} \\
\end{align}$
Now let us compare what we got with our law of exponents. We can see that there are two that we have to apply this law to. Let us look at the first variable, namely $c$ .
In the term ${{\left( {{c}^{2}} \right)}^{4}}$, we can clearly see that our $a$ is $c$ , our $m$ is $2$ and our $n$ is $4$.
And in the second term ${{\left( {{k}^{5}} \right)}^{4}}$, we can clearly see that our $a$ is $k$ , our $m$ is $5$ and our $n$ is $4$.
Now let us apply the law.
Upon applying we get the following :
$\begin{align}
  & \Rightarrow {{\left( {{c}^{2}}{{k}^{5}} \right)}^{4}} \\
 & \Rightarrow {{\left( {{c}^{2}} \right)}^{4}}{{\left( {{k}^{5}} \right)}^{4}} \\
 & \Rightarrow {{c}^{8}}{{k}^{20}} \\
\end{align}$
$\therefore $ Upon simplifying ${{\left( {{c}^{2}}{{k}^{5}} \right)}^{4}}$, we get ${{c}^{8}}{{k}^{20}}$.

Note: We should all the seven laws of exponents. We need to practice all those seven laws thoroughly so as to be able to solve a question related to this chapter quickly and accurately. We should be careful while substituting. This chapter can be clubbed with other chapters so as to form a question. Even graphs involving exponents can be asked. We should also be careful while solving as there is a huge amount of calculation errors.